Step |
Hyp |
Ref |
Expression |
1 |
|
bposlem2.1 |
|- ( ph -> N e. NN ) |
2 |
|
bposlem2.2 |
|- ( ph -> P e. Prime ) |
3 |
|
bposlem2.3 |
|- ( ph -> 2 < P ) |
4 |
|
bposlem2.4 |
|- ( ph -> ( ( 2 x. N ) / 3 ) < P ) |
5 |
|
bposlem2.5 |
|- ( ph -> P <_ N ) |
6 |
|
pcbcctr |
|- ( ( N e. NN /\ P e. Prime ) -> ( P pCnt ( ( 2 x. N ) _C N ) ) = sum_ k e. ( 1 ... ( 2 x. N ) ) ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) ) |
7 |
1 2 6
|
syl2anc |
|- ( ph -> ( P pCnt ( ( 2 x. N ) _C N ) ) = sum_ k e. ( 1 ... ( 2 x. N ) ) ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) ) |
8 |
|
elfznn |
|- ( k e. ( 1 ... ( 2 x. N ) ) -> k e. NN ) |
9 |
|
elnn1uz2 |
|- ( k e. NN <-> ( k = 1 \/ k e. ( ZZ>= ` 2 ) ) ) |
10 |
8 9
|
sylib |
|- ( k e. ( 1 ... ( 2 x. N ) ) -> ( k = 1 \/ k e. ( ZZ>= ` 2 ) ) ) |
11 |
|
oveq2 |
|- ( k = 1 -> ( P ^ k ) = ( P ^ 1 ) ) |
12 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
13 |
2 12
|
syl |
|- ( ph -> P e. NN ) |
14 |
13
|
nncnd |
|- ( ph -> P e. CC ) |
15 |
14
|
exp1d |
|- ( ph -> ( P ^ 1 ) = P ) |
16 |
11 15
|
sylan9eqr |
|- ( ( ph /\ k = 1 ) -> ( P ^ k ) = P ) |
17 |
16
|
oveq2d |
|- ( ( ph /\ k = 1 ) -> ( ( 2 x. N ) / ( P ^ k ) ) = ( ( 2 x. N ) / P ) ) |
18 |
17
|
fveq2d |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) = ( |_ ` ( ( 2 x. N ) / P ) ) ) |
19 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
20 |
14
|
mulid2d |
|- ( ph -> ( 1 x. P ) = P ) |
21 |
20 5
|
eqbrtrd |
|- ( ph -> ( 1 x. P ) <_ N ) |
22 |
|
1red |
|- ( ph -> 1 e. RR ) |
23 |
1
|
nnred |
|- ( ph -> N e. RR ) |
24 |
13
|
nnred |
|- ( ph -> P e. RR ) |
25 |
13
|
nngt0d |
|- ( ph -> 0 < P ) |
26 |
|
lemuldiv |
|- ( ( 1 e. RR /\ N e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( ( 1 x. P ) <_ N <-> 1 <_ ( N / P ) ) ) |
27 |
22 23 24 25 26
|
syl112anc |
|- ( ph -> ( ( 1 x. P ) <_ N <-> 1 <_ ( N / P ) ) ) |
28 |
21 27
|
mpbid |
|- ( ph -> 1 <_ ( N / P ) ) |
29 |
23 13
|
nndivred |
|- ( ph -> ( N / P ) e. RR ) |
30 |
|
1re |
|- 1 e. RR |
31 |
|
2re |
|- 2 e. RR |
32 |
|
2pos |
|- 0 < 2 |
33 |
31 32
|
pm3.2i |
|- ( 2 e. RR /\ 0 < 2 ) |
34 |
|
lemul2 |
|- ( ( 1 e. RR /\ ( N / P ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( 1 <_ ( N / P ) <-> ( 2 x. 1 ) <_ ( 2 x. ( N / P ) ) ) ) |
35 |
30 33 34
|
mp3an13 |
|- ( ( N / P ) e. RR -> ( 1 <_ ( N / P ) <-> ( 2 x. 1 ) <_ ( 2 x. ( N / P ) ) ) ) |
36 |
29 35
|
syl |
|- ( ph -> ( 1 <_ ( N / P ) <-> ( 2 x. 1 ) <_ ( 2 x. ( N / P ) ) ) ) |
37 |
28 36
|
mpbid |
|- ( ph -> ( 2 x. 1 ) <_ ( 2 x. ( N / P ) ) ) |
38 |
19 37
|
eqbrtrrid |
|- ( ph -> 2 <_ ( 2 x. ( N / P ) ) ) |
39 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
40 |
1
|
nncnd |
|- ( ph -> N e. CC ) |
41 |
13
|
nnne0d |
|- ( ph -> P =/= 0 ) |
42 |
39 40 14 41
|
divassd |
|- ( ph -> ( ( 2 x. N ) / P ) = ( 2 x. ( N / P ) ) ) |
43 |
38 42
|
breqtrrd |
|- ( ph -> 2 <_ ( ( 2 x. N ) / P ) ) |
44 |
|
2nn |
|- 2 e. NN |
45 |
|
nnmulcl |
|- ( ( 2 e. NN /\ N e. NN ) -> ( 2 x. N ) e. NN ) |
46 |
44 1 45
|
sylancr |
|- ( ph -> ( 2 x. N ) e. NN ) |
47 |
46
|
nnred |
|- ( ph -> ( 2 x. N ) e. RR ) |
48 |
|
3re |
|- 3 e. RR |
49 |
|
3pos |
|- 0 < 3 |
50 |
48 49
|
pm3.2i |
|- ( 3 e. RR /\ 0 < 3 ) |
51 |
|
ltdiv23 |
|- ( ( ( 2 x. N ) e. RR /\ ( 3 e. RR /\ 0 < 3 ) /\ ( P e. RR /\ 0 < P ) ) -> ( ( ( 2 x. N ) / 3 ) < P <-> ( ( 2 x. N ) / P ) < 3 ) ) |
52 |
50 51
|
mp3an2 |
|- ( ( ( 2 x. N ) e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( ( ( 2 x. N ) / 3 ) < P <-> ( ( 2 x. N ) / P ) < 3 ) ) |
53 |
47 24 25 52
|
syl12anc |
|- ( ph -> ( ( ( 2 x. N ) / 3 ) < P <-> ( ( 2 x. N ) / P ) < 3 ) ) |
54 |
4 53
|
mpbid |
|- ( ph -> ( ( 2 x. N ) / P ) < 3 ) |
55 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
56 |
54 55
|
breqtrdi |
|- ( ph -> ( ( 2 x. N ) / P ) < ( 2 + 1 ) ) |
57 |
47 13
|
nndivred |
|- ( ph -> ( ( 2 x. N ) / P ) e. RR ) |
58 |
|
2z |
|- 2 e. ZZ |
59 |
|
flbi |
|- ( ( ( ( 2 x. N ) / P ) e. RR /\ 2 e. ZZ ) -> ( ( |_ ` ( ( 2 x. N ) / P ) ) = 2 <-> ( 2 <_ ( ( 2 x. N ) / P ) /\ ( ( 2 x. N ) / P ) < ( 2 + 1 ) ) ) ) |
60 |
57 58 59
|
sylancl |
|- ( ph -> ( ( |_ ` ( ( 2 x. N ) / P ) ) = 2 <-> ( 2 <_ ( ( 2 x. N ) / P ) /\ ( ( 2 x. N ) / P ) < ( 2 + 1 ) ) ) ) |
61 |
43 56 60
|
mpbir2and |
|- ( ph -> ( |_ ` ( ( 2 x. N ) / P ) ) = 2 ) |
62 |
61
|
adantr |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( ( 2 x. N ) / P ) ) = 2 ) |
63 |
18 62
|
eqtrd |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) = 2 ) |
64 |
16
|
oveq2d |
|- ( ( ph /\ k = 1 ) -> ( N / ( P ^ k ) ) = ( N / P ) ) |
65 |
64
|
fveq2d |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( N / ( P ^ k ) ) ) = ( |_ ` ( N / P ) ) ) |
66 |
|
remulcl |
|- ( ( 2 e. RR /\ ( N / P ) e. RR ) -> ( 2 x. ( N / P ) ) e. RR ) |
67 |
31 29 66
|
sylancr |
|- ( ph -> ( 2 x. ( N / P ) ) e. RR ) |
68 |
48
|
a1i |
|- ( ph -> 3 e. RR ) |
69 |
|
4re |
|- 4 e. RR |
70 |
69
|
a1i |
|- ( ph -> 4 e. RR ) |
71 |
42 54
|
eqbrtrrd |
|- ( ph -> ( 2 x. ( N / P ) ) < 3 ) |
72 |
|
3lt4 |
|- 3 < 4 |
73 |
72
|
a1i |
|- ( ph -> 3 < 4 ) |
74 |
67 68 70 71 73
|
lttrd |
|- ( ph -> ( 2 x. ( N / P ) ) < 4 ) |
75 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
76 |
74 75
|
breqtrrdi |
|- ( ph -> ( 2 x. ( N / P ) ) < ( 2 x. 2 ) ) |
77 |
|
ltmul2 |
|- ( ( ( N / P ) e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( N / P ) < 2 <-> ( 2 x. ( N / P ) ) < ( 2 x. 2 ) ) ) |
78 |
31 33 77
|
mp3an23 |
|- ( ( N / P ) e. RR -> ( ( N / P ) < 2 <-> ( 2 x. ( N / P ) ) < ( 2 x. 2 ) ) ) |
79 |
29 78
|
syl |
|- ( ph -> ( ( N / P ) < 2 <-> ( 2 x. ( N / P ) ) < ( 2 x. 2 ) ) ) |
80 |
76 79
|
mpbird |
|- ( ph -> ( N / P ) < 2 ) |
81 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
82 |
80 81
|
breqtrdi |
|- ( ph -> ( N / P ) < ( 1 + 1 ) ) |
83 |
|
1z |
|- 1 e. ZZ |
84 |
|
flbi |
|- ( ( ( N / P ) e. RR /\ 1 e. ZZ ) -> ( ( |_ ` ( N / P ) ) = 1 <-> ( 1 <_ ( N / P ) /\ ( N / P ) < ( 1 + 1 ) ) ) ) |
85 |
29 83 84
|
sylancl |
|- ( ph -> ( ( |_ ` ( N / P ) ) = 1 <-> ( 1 <_ ( N / P ) /\ ( N / P ) < ( 1 + 1 ) ) ) ) |
86 |
28 82 85
|
mpbir2and |
|- ( ph -> ( |_ ` ( N / P ) ) = 1 ) |
87 |
86
|
adantr |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( N / P ) ) = 1 ) |
88 |
65 87
|
eqtrd |
|- ( ( ph /\ k = 1 ) -> ( |_ ` ( N / ( P ^ k ) ) ) = 1 ) |
89 |
88
|
oveq2d |
|- ( ( ph /\ k = 1 ) -> ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) = ( 2 x. 1 ) ) |
90 |
89 19
|
eqtrdi |
|- ( ( ph /\ k = 1 ) -> ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) = 2 ) |
91 |
63 90
|
oveq12d |
|- ( ( ph /\ k = 1 ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = ( 2 - 2 ) ) |
92 |
|
2cn |
|- 2 e. CC |
93 |
92
|
subidi |
|- ( 2 - 2 ) = 0 |
94 |
91 93
|
eqtrdi |
|- ( ( ph /\ k = 1 ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = 0 ) |
95 |
46
|
nnrpd |
|- ( ph -> ( 2 x. N ) e. RR+ ) |
96 |
95
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. N ) e. RR+ ) |
97 |
|
eluzge2nn0 |
|- ( k e. ( ZZ>= ` 2 ) -> k e. NN0 ) |
98 |
|
nnexpcl |
|- ( ( P e. NN /\ k e. NN0 ) -> ( P ^ k ) e. NN ) |
99 |
13 97 98
|
syl2an |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ k ) e. NN ) |
100 |
99
|
nnrpd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ k ) e. RR+ ) |
101 |
96 100
|
rpdivcld |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( 2 x. N ) / ( P ^ k ) ) e. RR+ ) |
102 |
101
|
rpge0d |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> 0 <_ ( ( 2 x. N ) / ( P ^ k ) ) ) |
103 |
47
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. N ) e. RR ) |
104 |
|
remulcl |
|- ( ( 3 e. RR /\ P e. RR ) -> ( 3 x. P ) e. RR ) |
105 |
48 24 104
|
sylancr |
|- ( ph -> ( 3 x. P ) e. RR ) |
106 |
105
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 3 x. P ) e. RR ) |
107 |
99
|
nnred |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ k ) e. RR ) |
108 |
|
ltdivmul |
|- ( ( ( 2 x. N ) e. RR /\ P e. RR /\ ( 3 e. RR /\ 0 < 3 ) ) -> ( ( ( 2 x. N ) / 3 ) < P <-> ( 2 x. N ) < ( 3 x. P ) ) ) |
109 |
50 108
|
mp3an3 |
|- ( ( ( 2 x. N ) e. RR /\ P e. RR ) -> ( ( ( 2 x. N ) / 3 ) < P <-> ( 2 x. N ) < ( 3 x. P ) ) ) |
110 |
47 24 109
|
syl2anc |
|- ( ph -> ( ( ( 2 x. N ) / 3 ) < P <-> ( 2 x. N ) < ( 3 x. P ) ) ) |
111 |
4 110
|
mpbid |
|- ( ph -> ( 2 x. N ) < ( 3 x. P ) ) |
112 |
111
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. N ) < ( 3 x. P ) ) |
113 |
24 24
|
remulcld |
|- ( ph -> ( P x. P ) e. RR ) |
114 |
113
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P x. P ) e. RR ) |
115 |
|
nnltp1le |
|- ( ( 2 e. NN /\ P e. NN ) -> ( 2 < P <-> ( 2 + 1 ) <_ P ) ) |
116 |
44 13 115
|
sylancr |
|- ( ph -> ( 2 < P <-> ( 2 + 1 ) <_ P ) ) |
117 |
3 116
|
mpbid |
|- ( ph -> ( 2 + 1 ) <_ P ) |
118 |
55 117
|
eqbrtrid |
|- ( ph -> 3 <_ P ) |
119 |
|
lemul1 |
|- ( ( 3 e. RR /\ P e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( 3 <_ P <-> ( 3 x. P ) <_ ( P x. P ) ) ) |
120 |
48 119
|
mp3an1 |
|- ( ( P e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( 3 <_ P <-> ( 3 x. P ) <_ ( P x. P ) ) ) |
121 |
24 24 25 120
|
syl12anc |
|- ( ph -> ( 3 <_ P <-> ( 3 x. P ) <_ ( P x. P ) ) ) |
122 |
118 121
|
mpbid |
|- ( ph -> ( 3 x. P ) <_ ( P x. P ) ) |
123 |
122
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 3 x. P ) <_ ( P x. P ) ) |
124 |
14
|
sqvald |
|- ( ph -> ( P ^ 2 ) = ( P x. P ) ) |
125 |
124
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ 2 ) = ( P x. P ) ) |
126 |
24
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> P e. RR ) |
127 |
13
|
nnge1d |
|- ( ph -> 1 <_ P ) |
128 |
127
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> 1 <_ P ) |
129 |
|
simpr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> k e. ( ZZ>= ` 2 ) ) |
130 |
126 128 129
|
leexp2ad |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ 2 ) <_ ( P ^ k ) ) |
131 |
125 130
|
eqbrtrrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P x. P ) <_ ( P ^ k ) ) |
132 |
106 114 107 123 131
|
letrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 3 x. P ) <_ ( P ^ k ) ) |
133 |
103 106 107 112 132
|
ltletrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. N ) < ( P ^ k ) ) |
134 |
99
|
nncnd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( P ^ k ) e. CC ) |
135 |
134
|
mulid1d |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( P ^ k ) x. 1 ) = ( P ^ k ) ) |
136 |
133 135
|
breqtrrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. N ) < ( ( P ^ k ) x. 1 ) ) |
137 |
|
1red |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> 1 e. RR ) |
138 |
103 137 100
|
ltdivmuld |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( ( 2 x. N ) / ( P ^ k ) ) < 1 <-> ( 2 x. N ) < ( ( P ^ k ) x. 1 ) ) ) |
139 |
136 138
|
mpbird |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( 2 x. N ) / ( P ^ k ) ) < 1 ) |
140 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
141 |
139 140
|
breqtrdi |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( 2 x. N ) / ( P ^ k ) ) < ( 0 + 1 ) ) |
142 |
101
|
rpred |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( 2 x. N ) / ( P ^ k ) ) e. RR ) |
143 |
|
0z |
|- 0 e. ZZ |
144 |
|
flbi |
|- ( ( ( ( 2 x. N ) / ( P ^ k ) ) e. RR /\ 0 e. ZZ ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) = 0 <-> ( 0 <_ ( ( 2 x. N ) / ( P ^ k ) ) /\ ( ( 2 x. N ) / ( P ^ k ) ) < ( 0 + 1 ) ) ) ) |
145 |
142 143 144
|
sylancl |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) = 0 <-> ( 0 <_ ( ( 2 x. N ) / ( P ^ k ) ) /\ ( ( 2 x. N ) / ( P ^ k ) ) < ( 0 + 1 ) ) ) ) |
146 |
102 141 145
|
mpbir2and |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) = 0 ) |
147 |
1
|
nnrpd |
|- ( ph -> N e. RR+ ) |
148 |
147
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> N e. RR+ ) |
149 |
148 100
|
rpdivcld |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( N / ( P ^ k ) ) e. RR+ ) |
150 |
149
|
rpge0d |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> 0 <_ ( N / ( P ^ k ) ) ) |
151 |
23
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> N e. RR ) |
152 |
23 147
|
ltaddrpd |
|- ( ph -> N < ( N + N ) ) |
153 |
40
|
2timesd |
|- ( ph -> ( 2 x. N ) = ( N + N ) ) |
154 |
152 153
|
breqtrrd |
|- ( ph -> N < ( 2 x. N ) ) |
155 |
154
|
adantr |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> N < ( 2 x. N ) ) |
156 |
151 103 107 155 133
|
lttrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> N < ( P ^ k ) ) |
157 |
156 135
|
breqtrrd |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> N < ( ( P ^ k ) x. 1 ) ) |
158 |
151 137 100
|
ltdivmuld |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( N / ( P ^ k ) ) < 1 <-> N < ( ( P ^ k ) x. 1 ) ) ) |
159 |
157 158
|
mpbird |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( N / ( P ^ k ) ) < 1 ) |
160 |
159 140
|
breqtrdi |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( N / ( P ^ k ) ) < ( 0 + 1 ) ) |
161 |
149
|
rpred |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( N / ( P ^ k ) ) e. RR ) |
162 |
|
flbi |
|- ( ( ( N / ( P ^ k ) ) e. RR /\ 0 e. ZZ ) -> ( ( |_ ` ( N / ( P ^ k ) ) ) = 0 <-> ( 0 <_ ( N / ( P ^ k ) ) /\ ( N / ( P ^ k ) ) < ( 0 + 1 ) ) ) ) |
163 |
161 143 162
|
sylancl |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( |_ ` ( N / ( P ^ k ) ) ) = 0 <-> ( 0 <_ ( N / ( P ^ k ) ) /\ ( N / ( P ^ k ) ) < ( 0 + 1 ) ) ) ) |
164 |
150 160 163
|
mpbir2and |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( |_ ` ( N / ( P ^ k ) ) ) = 0 ) |
165 |
164
|
oveq2d |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) = ( 2 x. 0 ) ) |
166 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
167 |
165 166
|
eqtrdi |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) = 0 ) |
168 |
146 167
|
oveq12d |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = ( 0 - 0 ) ) |
169 |
|
0m0e0 |
|- ( 0 - 0 ) = 0 |
170 |
168 169
|
eqtrdi |
|- ( ( ph /\ k e. ( ZZ>= ` 2 ) ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = 0 ) |
171 |
94 170
|
jaodan |
|- ( ( ph /\ ( k = 1 \/ k e. ( ZZ>= ` 2 ) ) ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = 0 ) |
172 |
10 171
|
sylan2 |
|- ( ( ph /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = 0 ) |
173 |
172
|
sumeq2dv |
|- ( ph -> sum_ k e. ( 1 ... ( 2 x. N ) ) ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = sum_ k e. ( 1 ... ( 2 x. N ) ) 0 ) |
174 |
|
fzfid |
|- ( ph -> ( 1 ... ( 2 x. N ) ) e. Fin ) |
175 |
|
sumz |
|- ( ( ( 1 ... ( 2 x. N ) ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... ( 2 x. N ) ) e. Fin ) -> sum_ k e. ( 1 ... ( 2 x. N ) ) 0 = 0 ) |
176 |
175
|
olcs |
|- ( ( 1 ... ( 2 x. N ) ) e. Fin -> sum_ k e. ( 1 ... ( 2 x. N ) ) 0 = 0 ) |
177 |
174 176
|
syl |
|- ( ph -> sum_ k e. ( 1 ... ( 2 x. N ) ) 0 = 0 ) |
178 |
173 177
|
eqtrd |
|- ( ph -> sum_ k e. ( 1 ... ( 2 x. N ) ) ( ( |_ ` ( ( 2 x. N ) / ( P ^ k ) ) ) - ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) ) = 0 ) |
179 |
7 178
|
eqtrd |
|- ( ph -> ( P pCnt ( ( 2 x. N ) _C N ) ) = 0 ) |